Throughout this paper, for a ring \(R\) , the symbol \(R^*\)  denotes the group of all units in \(R\) . For a non-empty subset \(S\)  of \(D\) , we denote by \(F[S]\)  and \(F(S)\)  the subring and the division subring of \(D\)  respectively, generated by the set \(F\cup S\) , where \(F=Z(D)\)  is the center of \(D\) . We say that a subgroup \(G\)  of \(D^*\)  is absolutely irreducible if \(F[G]=D\) . Given a group \(G\) , a subset \(S\)  and a subgroup \(H\)  of \(G\) , we denote by \(Z(G)\) , \(G^{\prime }\) , \(C_G(S)\)  and \(H_G=\bigcap _{x\in G} xHx^{-1}\)  the center of \(G\) , the derived group of \(G\) , the centralizer of \(S\)  in \(G\)  and the core of \(H\)  in \(G\)  respectively. If \(x\in G\)  and \(H\)  is a subgroup of \(G\) , then we denote by \(x^H\)  the set of all elements \(x^h=hxh^{-1}, h\in H\) . If \(x^G\)  is finite, then we say that \(x\)  is an \(FC\) -element of \(G\) . The set of all \(FC\) -elements of \(G\)  is called the \(FC\) -center of \(G\) . If \(x, y\in G\)  then \([x,y]=xyx^{-1}y^{-1}\)  and \([H,K]\)  is the subgroup of \(G\)  generated by all elements \([h, k], h\in H,\,k\in K\) . We say that an element \(x\in D\)  is radical over \(F\)  if there exists a positive integer \(n(x)\)  depending on \(x\)  such that \(x^{n(x)}\in F\) . A non-empty subset \(S\)  of \(D\)  is radical over \(F\)  if every element of \(S\)  is radical over \(F\) . We write \(H\le G\)  and \(H<G\)  if \(H\)  is a subgroup and proper subgroup of \(G\)  respectively. All other notation and symbols in this paper are standard and one can find, for example, in [1]}, [2]}, [3]}, [4]}, [5]}.
